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Proof step 30 from proof replayorphans-prepost (previous step)

key0 ∈ (ri0 \ ks0) \ datakeys(ks0, ri0) ↔ key0 ∈ ri1 ⊖ ro1, ((ri0 \ ks0) \ datakeys(ks0, ri0))[key0] = (ri1 ⊖ ro1)[key0], ks0.inodes?, ro1.inodes?, key ∈ ro1, key ∈ ks0, ks0 ⊆ ro1, ∀ key. (key ∈ (ri0 \ ks0) \ datakeys(ks0, ri0) ↔ key ∈ ri1 ⊖ ro1) ∧ ((ri0 \ ks0) \ datakeys(ks0, ri0))[key] = (ri1 ⊖ ro1)[key] ⊦ # ro1 = 0, # ks0 = 0, ks0 = ∅, ro1 = ∅, key0 ∈ ri0 ∧ ¬ key0 ∈ ks0 ∧ ¬ key0.data?, (key0 ∈ ((ri0 \ datakeys(key, ri0)) -- key) ⊖ (ks0 -- key) ↔ key0 ∈ ri1 ⊖ ro1) ∧ (((ri0 \ datakeys(key, ri0)) -- key) ⊖ (ks0 -- key))[key0] = (ri1 ⊖ ro1)[key0]

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 31:
(key0 ∈ ri0 ∧ key0.data? ∧ inodekey(key0.ino) ∈ ks0 ⊃ (∅)[key0];(key0 ∈ ks0 ⊃ (∅)[key0];ri0[key0])) = (ri1 ⊖ ro1)[key0], key0 ∈ ri0 ∧ ¬ key0 ∈ ks0 ∧ (¬ key0.data? ∨ ¬ inodekey(key0.ino) ∈ ks0) ↔ key0 ∈ ri1 ⊖ ro1, ro1.inodes?, ks0.inodes?, key ∈ ks0, key ∈ ro1, ks0 ⊆ ro1, ∀ key1. (key1 ∈ ri0 ∧ ¬ key1 ∈ ks0 ∧ (¬ key1.data? ∨ ¬ inodekey(key1.ino) ∈ ks0) ↔ key1 ∈ ri1 ⊖ ro1) ∧ (key1 ∈ ri0 ∧ key1.data? ∧ inodekey(key1.ino) ∈ ks0 ⊃ (∅)[key1];(key1 ∈ ks0 ⊃ (∅)[key1];ri0[key1])) = (ri1 ⊖ ro1)[key1] ⊦ # ks0 = 0, # ro1 = 0, ks0 = ∅, ro1 = ∅, (key0 ∈ ((ri0 \ datakeys(key, ri0)) -- key) ⊖ (ks0 -- key) ↔ key0 ∈ ri1 ⊖ ro1) ∧ (((ri0 \ datakeys(key, ri0)) -- key) ⊖ (ks0 -- key))[key0] = (ri1 ⊖ ro1)[key0], key0 ∈ ri0 ∧ ¬ key0 ∈ ks0 ∧ ¬ key0.data?

Rule argument

Additional information

Used simplifier rules: at-diff-ri: ⊦ (ri \ ks)[key] = (key ∈ ks ⊃ (∅)[key];ri[key]);% ⊦ key ∈ ri \ ks ↔ key ∈ ri ∧ ¬ key ∈ ks; (ls)%⊦ key ∈ datakeys(ks, ri) ↔ key ∈ ri ∧ key.data? ∧ inodekey(key.ino) ∈ ks; (s)